You can trace over all of the diagonals of a pentagon without
lifting your pencil and without going over any more than once. Can
the same thing be done with a hexagon or with a heptagon?

Hamilton's Puzzle

Stage: 2 Challenge Level:

In 1859, the Irish mathematician Sir William Rowan Hamilton devised a
puzzle with a regular dodecahedron made of wood. Here is a dodecahedron:

He labelled each of the vertices with the name of an important city. The
challenge was to find a route along the edges of the dodecahedron which
visited every city exactly once and returned to the start.

Here is a graph which represents the dodecahedron. Can you see how each
of the 20 vertices, 30 edges and 12 pentagonal faces is represented in
the graph?

I start my journey in Rio de Janeiro and visit all the cities as
Hamilton described, passing through Canberra before Madrid, and then
returning to Rio. What route could I have taken?

Can you find any other ways of making this journey?

Here is a simpler network of countries:

How many different ways are there of visiting each of these countries
once and only once, beginning and ending at Australia?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.